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frank0520

  • one year ago

Let F be the vector field F=(13x^2 y+3y^3 -y)i -12x^3 j. Find the maximum value of ∫F•dr where c is positively oriented simple closed curve.

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  1. ganeshie8
    • one year ago
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    You may start with green's thm I think

  2. ganeshie8
    • one year ago
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    @dan815

  3. ganeshie8
    • one year ago
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    \[\oint F.dr = \iint\limits_{R} \text{curl}(F)~dA = \iint\limits_{R} -49x^2-9y^2+1~dA \]

  4. ganeshie8
    • one year ago
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    Easy to see that the integrand stays positive in the region \(49x^2+9y^2 \lt 1\), so should we pick this as region of integration ?

  5. frank0520
    • one year ago
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    Here is a picture of the question.

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  6. ganeshie8
    • one year ago
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    Then we're correct, go ahead and evaluate the integral over that ellipse

  7. ganeshie8
    • one year ago
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    \[max(\oint F.dr) ~= ~\iint\limits_{49x^2+9y^2\lt 1}~~1-49x^2-9y^2~dA\]

  8. ganeshie8
    • one year ago
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    you will need to use change of variables

  9. ganeshie8
    • one year ago
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    I'm getting \(\dfrac{21\pi}{2}\) http://www.wolframalpha.com/input/?i=21*%5Cint_0%5E%282pi%29%5Cint_0%5E1+%281-r%5E2%29*r+dr+d%5Ctheta

  10. frank0520
    • one year ago
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    looks correct to me, I got \[\frac{ 42\pi }{ 4 }\]

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